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Generalized ulam-hyers stability of C*-Ternary algebra n-Homomorphisms for a functional equation

Journal of Inequalities and Applications20112011:30

https://doi.org/10.1186/1029-242X-2011-30

Received: 15 December 2010

Accepted: 9 August 2011

Published: 9 August 2011

Abstract

In this article, we investigate the Ulam-Hyers stability of C*-ternary algebra n-homomorphisms for the functional equation:

in C*-ternary algebras.

2000 Mathematics Subject Classification: Primary 39B82; 46B03; 47Jxx.

Keywords

n-homomorphismsC*-ternary algebra

1. Introduction and preliminaries

Ternary algebraic operations were considered in the nineteenth century by several mathematicians, such as Cayley [1] who introduced the notion of cubic matrix, which, in turn, was generalized by Kapranov et al. [2]. The simplest example of such nontrivial ternary operation is given by the following composition rule:
Ternary structures and their generalization, the so-called n-array structures, raise certain hopes in view of their applications in physics. Some significant physical applications are as follows (see [3]):
  1. (1)
    The algebra of nonions generated by two matrices
     
was introduced by Sylvester as a ternary analog of Hamilton's quaternions [4].
  1. (2)

    The quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechnics is based on such structures [5].

     

There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics, supersymmetric theory, and Yang-Baxter equation [4, 6].

A C*-ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) α [x, y, z] of A3 into A, which is -linear in the outer variables, conjugate -linear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies ||[x, y, z]|| ≤ ||x|| · ||y|| · ||z|| and ||[x, x, x]|| = ||x||3 (see [7, 8]). Every left Hilbert C*-module is a C*-ternary algebra via the ternary product [x, y, z] := 〈x, yz.

If a C*-ternary algebra (A,[·, ·, ·]) has an identity, i.e., an element e A such that x = [x, e, e] = [e, e, x] for all x A, then it is customary to verify that A, endowed with x y := [x, e, y] and x* := [e, x, e], is a unital C*-algebra. Conversely, if (A, ) is a unital C*-algebra, then [x, y, z] := x y* z makes A into a C*-ternary algebra.

Let A and B be C*-ternary algebras. A -linear mapping H : AB is called a C*-ternary algebra homomorphism if

for all x, y, z A.

Definition. Let A and B be C*-ternary algebras. A multilinear mapping H : A n B over is called a C*-ternary algebra n-homomorphism if it satisfies

for all x1, y1, z1, · · ·, x n , y n , z n A.

In 1940, Ulam [9] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms:

We are given a group G and a metric group G' with metric ρ(·, ·). Given ε > 0, does there exist a δ > 0 such that if f : GG' satisffies ρ(f(xy), f(x) f(y)) < δ for all x, y G, then a homomorphism h : GG' exists with ρ(f(x), h(x)) < ε for all x G ?

In 1941, Hyers [10] gave the first partial solution to Ulam's question for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. Then, Aoki [11] and Bourgin [12] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [13] generalized the theorem of Hyers [10] by considering the stability problem with unbounded Cauchy differences. In 1991, Gajda [14], following the same approach as that by Rassias [13], gave an affirmative solution to this question for p > 1. It was shown by Gajda [14] as well as by Rassias and Šemrl [15], that one cannot prove a Rassias-type theorem when p = 1. Găvruta [16] obtained the generalized result of Rassias's theorem which allows the Cauchy difference to be controlled by a general unbounded function. During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings, for example, Cauchy-Jensen mappings, k-additive mappings, invariant means, multiplicative mappings, bounded n th differences, convex functions, generalized orthogonality mappings, Euler-Lagrange functional equations, differential equations, and Navier-Stokes equations. The instability of characteristic flows of solutions of partial differential equations is related to the Ulam's stability of functional equations [17]-[24]. On the other hand, the authors [25], Park [20] and Kim [26] have contributed studies in respect of the stability problem of ternary homomorphisms and ternary derivations.

2. Solution and stability

Let X and Y be real or complex vector spaces and n ≥ 2 an integer. For a mapping f : X n Y, consider the functional equation:
(2.1)
The above functional equation is rewritten as
(2.2)
where

We solve the general problem in vector spaces for the n-additive mappings satisfying (2.1).

Theorem 2.1. A mapping f : X n Y satisffies the equation (2.1) if and only if the mapping f is n-additive.

Proof. Assume that f satisfies (2.1). Letting x1 = y1 = · · · = x n = y n = 0 in (2.2), we get f(0, · · ·, 0) = 0. Letting y1 = x2 = y2 = · · · = x n = y n = 0 in (2.2), we have
for all x1 X. Similarly, we get
for all x2, · · ·, x n X. Setting y1 = y2 = 0 and x3 = y3 = · · · = x n = y n = 0 in (2.2), we have

for all x1, x2 X. Similarly, we get f(x1, 0, x3, 0, · · ·, 0) = · · · = f(0, · · ·, 0, xn-1, x n ) = 0 for all x1, · · ·, x n X.

Continuing this process, we obtain that f(x1, · · ·, x n ) = 0 for all x1, · · ·, x n X with x i = 0 for some i = 1, · · ·, n. Letting y2 = · · · = y n = 0 in (2.2), we get the additivity in the first variable. Similarly, the additivities in the remaining variables hold.

The converse is obvious. □

We investigate the generalized Ulam's stability in C*-ternary algebras for the n-additive mappings satisfying (2.1).

Lemma 2.2. Let X and Y be complex vector spaces and let f : X n Y be a n-additive mapping such that

for all and all x1, · · ·, x n X, then f is n-linear over .

Proof. Since f is n-additive, we get for all x1, · · ·, x n X. Now let and M be an integer greater than 2(|σ1| + · · · + |σ n |). Since , there is such that . Now
for some . Thus, we have

for all x1, · · ·, x n X. Hence, the mapping f : X n Y is n-linear over . □

Using the above lemma, one can obtain the following result.

Theorem 2.3. Let X and Y be complex vector spaces and let f : X n Y be a mapping such that
(2.3)

for all and all x1,1, x2,1, · · ·, x1,n, x2,n X. Then, f is n-linear over .

Proof. Letting λ1 = · · · = λ n = 1, by Theorem 1.1, f is n-additive. Letting x2,1 = · · · = x2,n= 0 in (2.3), we get f(λ1x1, · · ·, λ n x n ) = λ1 · · · λ n f(x1, · · ·, x n ) for all and all x1, · · ·, x n X. Hence, by Lemma 2.2, the mapping f is n-linear over . □

From now on, assume that A is a C*-ternary algebra with norm || · || A and that B is a C*-ternary algebra with norm || · || B .

For a given mapping f : A n B, we define

for all and all x1,1, x2,1, · · ·, x1,n, x2,n A.

We prove the generalized Ulam-Hyers stability of homomorphisms in C*-ternary algebras for the functional equation
Theorem 2.4. Let p1, ···, p n (0, ∞) with and θ (0, ∞), and let f : A n B be a mapping such that
(2.4)
and
(2.5)
for all and all x1, y1, z1, · · ·, x n , y n , z n A. Then, there exists a unique C*-ternary algebra n-homomorphism H : A n B such that
(2.6)

for all x1, · · ·, x n A.

Proof. Letting λ1 = · · · = λ n = 1, y1 = x1, · · ·, y n = x n in (2.4), we gain
(2.7)
for all x1, · · ·, x n A. Thus, we have
for all x1, · · ·, x n A and all j . For given integer l, m(0 ≤ l < m), we obtain that
(2.8)
for all x1, · · ·, x n A. Since , the sequence
is a Cauchy sequence for all x1, ···, x n A. Since B is complete, the sequence converges for all x1, · · ·, x n A. Define H : A n B by
for all x1, · · ·, x n A. Letting l = 0 and taking m → ∞ in (2.8), one can obtain the inequality (2.6). By (2.4), we see that

for all x1, y1, · · ·, x n , y n A and all s. Since , letting s → ∞ in the above inequality, H satisfies (2.1). By Theorem 2.1, H is n-additive.

Letting y1 = x1, · · ·, y n = x n in (2.4), we gain
for all and all x1, · · ·, x n A. Thus we have
for all , all x1, · · ·, x n A and all m . Hence, we get
for all x1, · · ·, x n A and all m , and one can show that
for all , all x1, · · ·, x n A and all m . Hence,
for all , all x1, · · ·, x n A and all m . Since , we have
as m → ∞ for all and all x1, · · ·, x n A. Hence
for all and all x1, ···, x n A. From Lemma 2.2, the mapping H : A n B is n-linear over . It follows from (2.5) that
for all x1, y1, z1, ···, x n , y n , z n A. So

for all x1, y1, z1, ···, x n , y n , z n A.

Now, let T : A n B be another n-additive mapping satisfying (2.6). Then, we have

which tends to zero as m → ∞ for all x1, ···, x n A. Hence, we can conclude that H(x1, ···, x n ) = T(x1, ···, x n ) for all x1, ···, x n A. This proves the uniqueness of H.

Thus, the mapping H : AB is a unique C*-ternary algebra n-homomorphism satisfying (2.6). □

Letting p1 = ··· = p n = 0 and θ = ε in Theorem 2.4, we obtain the Ulam-Hyers stability for the n-additive functional equation (2.1).

Corollary 2.5. Let ε (0, ∞) and let f : A n B be a mapping satisfying
and
for all and all x1, y1, z1 ···, x n , y n , z n A. Then, there exists a unique C*-ternary algebra n-homomorphism H : An → B such that

for all x1, ···, x n A.

Example 2.6. We present the following counterexample modiffied by the well-known counterexample of Z. Gajda[14]for the functional equation (2.1). Set θ > 0 and let .

Deffine a function f : n by
for all x1, ···, x n , where ϕ μ : n is the function given by
for all x1, ···, x n . Deffine another function g : by

for all x .

For all m and all , we assert that
(2.9)
It was proved in[14]that
for all x, y , that is, (2.9) holds for m = 1. For a ffixed k , assume that (2.9) holds for m = k. Then, we have

for all , that is, (2.9) holds for m = k + 1. Hence, (2.9) holds for all m .

Note that
(2.10)
for all x1, ···, x n . By the inequality (2.9) and the above equality, we see that
for all x1, y1, ···, x n , y n . However, we observe from[14]that
and so
Thus,
where h: n is the function given by

for all x1, ···, x n . Hence, the function f is a counterexample for the singular case of Theorem 2.4.

Theorem 2.7. Let p (0,n) and θ (0, ∞), and let f : A n B be a mapping such that
(2.11)
and
(2.12)
for all and all x1, y1, z1 ···, x n , y n , z n A. Then, there exists a unique C*-ternary algebra n-homomorphism H : A n B such that

for all x1, ···, x n A.

Proof. The proof is similar to the proof of Theorem 2.4. □

Example 2.8. We present the following counterexample modiffied by the well-known counterexample of Z. Gajda[14]for the functional equation (2.1). Set θ > 0 and let

Let f : n and g : be the same as in Example 2.6. By the same argument as in Example 2.6, for all m and all , one can obtain that g satisffies the inequality:
By the equality (2.10) and the above inequality, we see that
for all x1, y1, ···, x n , y n . For each x1, y1, ···, x n , y n , let M(x1, y1, ···, x n , y n ) := max{|x1|, |y1|, ···, |x n |, |y n |}. We have
for all x1, y1, ···, x n , y n . Thus we have

for all x1, y1, ···, x n , y n . By the same reason as for Example 2.6, the function f is a counterexample for the singular case p = n of Theorem 2.7.

Theorem 2.9. Let p1, ···, p n (0, ∞) with , s (0, n) and θ, η (0, ∞), and let f: A n B be a mapping such that
(2.13)
and
(2.14)
for all and all x1,1, x2,1, x3,1, ···, x1, n, x2, n, x3, n A. Then, there exists a unique C*-ternary algebra n-homomorphism H: A n B such that

for all x1, ···, x n A.

Proof. The proof is similar to the proof of Theorem 2.4. □

Theorem 2.10. Let p1, ···, p n (0, ∞) with and θ (0, ∞), and let f : A n B be a mapping satisfying (2.4) (2.5). Then, there exists a unique C*-ternary algebra n-homomorphism H : A n B such that
(2.15)

for all x1, ···, x n A.

Proof. It follows from (2.7) that
for all x1, ···, x n A. Hence,
(2.16)

for all nonnegative integers m and l with m > l and all x1, ···, x n A. It follows from (2.16) that the sequence is a Cauchy sequence for all x1, ···, x n A. Since B is complete, the sequence converges. Hence, one can define the mapping H : A n B by for all x1, ···, x n A. Moreover, letting l = 0 and passing the limit m → ∞ in (2.16), we get (2.15).

The remainder of the proof is similar to the proof of Theorem 2.4. □

Theorem 2.11. Let p (3n, ∞) and θ (0, ∞), and let f: A n B be a mapping satisfying (2.11) (2.7), and f(0, ···, 0) = 0. Then, there exists a unique C*-ternary algebra n-homomorphism H : A n B such that

for all x1, ···, x n A.

Proof. The proof is similar to that of Theorem 2.10. □

Theorem 2.12. Let p1, ···, p n (0, ∞) with , s (n, ∞) and θ, η (0, ∞), and let f: A n B be a mapping such that (2.13), (2.14), and f(0, ···, 0) = 0. Then, there exists a unique C*-ternary algebra n-homomorphism H: A n B such that

for all x1, ···, x n A.

Proof. The proof is similar to that of Theorem 2.10.

Declarations

Acknowledgements

The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.

Authors’ Affiliations

(1)
Department of Mathematics Education, Mokwon University, Daejeon, Republic of Korea
(2)
Humanitas College, Kyung Hee University, Yongin, Republic of Korea

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Copyright

© Park and Bae; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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